To construct a polynomial with real coefficients, we need to use the given information:

• Degree 4: The polynomial will have 4 roots (including multiplicities).
• Zeros:
  • $3 - 4i$: Since the coefficients must be real, the complex conjugate $3 + 4i$ is also a zero.
  • $-2$ with multiplicity 2: This means the factor $(x + 2)$ appears twice.

Step-by-step construction:

1. Form factors for the zeros:

   • For the complex conjugate pair $3 - 4i$ and $3 + 4i$: $\left( x - (3 - 4i) \right) \left( x - (3 + 4i) \right)$ Use the difference of squares formula: $\left( x - 3 \right)^2 + 16 = x^2 - 6x + 25$

   • For the zero $-2$ with multiplicity 2: $(x + 2)^2 = x^2 + 4x + 4$

2. Form the polynomial: Multiply the two factors: $f(x) = \left( x^2 - 6x + 25 \right) \left( x^2 + 4x + 4 \right)$

3. Expand the product: Use distribution (multiplying every term in one factor by every term in the other):

   $f(x) = x^4 + 4x^3 + 4x^2 - 6x^3 - 24x^2 - 24x + 25x^2 + 100x + 100$

   Now, combine like terms: $f(x) = x^4 - 2x^3 + 5x^2 + 76x + 100$

Thus, the polynomial is: $f(x) = x^4 - 2x^3 + 5x^2 + 76x + 100$

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Do you have any questions or need more details?

Here are 5 related questions:

1. How do you find the complex conjugate of a given complex number?
2. What is the difference of squares formula and when is it used?
3. How do multiplicities affect the shape of a polynomial’s graph?
4. What is the fundamental theorem of algebra?
5. How do you determine the degree of a polynomial?

Tip: Multiplying complex conjugate pairs always gives a quadratic expression with real coefficients.